Approximation of the interface condition for stochastic Stefan-type problems

TL;DR

This paper studies the approximation of interface conditions in stochastic Stefan problems by volume imbalances, analyzing convergence of solutions in a complex stochastic evolution framework with non-linear coefficients.

Contribution

It introduces a novel approach to approximate interface conditions in stochastic Stefan problems and proves convergence of solutions under less restrictive coefficient conditions.

Findings

Established convergence of solutions with volume-based interface approximations.

Analyzed stochastic evolution equations with non-linear, non-globally Lipschitz coefficients.

Provided continuity properties of the mild solution map in complex stochastic settings.

Abstract

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.